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HMMT 十一月 2008 · 冲刺赛 · 第 17 题

HMMT November 2008 — Guts Round — Problem 17

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 10 ] Determine the last two digits of 17 , written in base 10. 6
解析

英文解析

  1. [ 10 ] Determine the last two digits of 17 , written in base 10.
    Answer: 77 We are asked to find the remainder when 17 is divided by 100. Write the power as 17
    (7 + 10) and expand with the binomial theorem:17
    17 17 16
    (7 + 10) = 7 + 17 · 7 · 10 + . . .
    We can ignore terms with more than one factor of 10 because these terms are divisible by 100, soadding them does not change the last two digits. Now, 7 = 2401, which has remainder 1 mod 100, so 4
    17 16
    7 has last two digits 07 and 7 · 10 has last two digits 70. We add these together. 6